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Empirical derivation, without relativity
Some claim that the best empirical verification of '''E=mc<sup>2</sup>''' was done in 2005 by Simon Rainville et al., as published in ''Nature'' (which is not a leading physics journal).<ref>[http://www.nature.com/nature/journal/v438/n7071/full/4381096a.html Nature 438, 1096-1097 (22 December 2005)] doi:10.1038/4381096a; Published online 21 December 2005</ref> The authors state in their article in ''Nature'' magazine that "Einstein's relationship is separately confirmed in two tests, which yield a combined result of 1−Δmc²/E=(−1.4±4.4)×10<sup>−7</sup>, indicating that it holds to a level of at least 0.00004%. To our knowledge, this is the most precise direct test of the famous equation yet described."
Some claim that the best empirical verification of '''E=mc<sup>2</sup>''' was done in 2005 by Simon Rainville et al., as published in ''Nature'' (which is not a leading physics journal).<ref>[http://www.nature.com/nature/journal/v438/n7071/full/4381096a.html Nature 438, 1096-1097 (22 December 2005)] doi:10.1038/4381096a; Published online 21 December 2005</ref> The authors state in their article in ''Nature'' magazine that "Einstein's relationship is separately confirmed in two tests, which yield a combined result of 1−Δmc²/E=(−1.4±4.4)×10<sup>−7</sup>, indicating that it holds to a level of at least 0.00004%. To our knowledge, this is the most precise direct test of the famous equation yet described."
==An Isolated Example -- Nuclear Fission of Uranium==
==A Famous Example -- Nuclear Fission of Uranium==
For most types of physical interactions, the masses of the initial reactants and of the final products match so closely that it is essentially impossible to measure any difference. But for nuclear reactions, the difference is measurable. That difference is related to the energy absorbed or released, described by the equation E=mc². (The equation applies to '''all''' interactions; the fact that nuclear interactions are the only ones for which the mass difference is measurable has led people to believe, wrongly, that E=mc² applies only to nuclear interactions.)
For most types of physical interactions, the masses of the initial reactants and of the final products match so closely that it is essentially impossible to measure any difference. But for nuclear reactions, the difference is measurable. That difference is related to the energy absorbed or released, described by the equation E=mc². (The equation applies to '''all''' interactions; the fact that nuclear interactions are the only ones for which the mass difference is measurable has led people to believe, wrongly, that E=mc² applies only to nuclear interactions.)
The insight that the conversion from Uranium to Barium was caused by complete fission of the atom was made by Lise Meitner in December, 1938. She had the approximate "mass defect" quantities memorized, and so she worked out in her head, using the E=mc² equation, that there would be this enormous release of energy. This release was observed shortly thereafter, and the result is nuclear power and nuclear weapons.
The insight that the conversion from Uranium to Barium was caused by complete fission of the atom was made by Lise Meitner in December, 1938. She had the approximate "mass defect" quantities memorized, and so she worked out in her head, using the E=mc² equation, that there would be this enormous release of energy. This release was observed shortly thereafter, and the result is nuclear power and nuclear weapons.
==A Counterexample: Speed of Extremely Energetic Neutrinos==
==A Topical Example: Speed of Extremely Energetic Neutrinos==
Here is another example of the use of this formula in physics calculations. Recently there has been quite a controversy over whether neutrinos were observed traveling at a speed faster than light. Relativity doesn't allow that, and, since neutrinos have nonzero (but incredibly tiny) mass, they aren't even supposed to travel ''at'' the speed of light. This very issue came up on the [[Talk:Main_Page#Neutrinos]]. The speeds under discussion were calculated by the use of E=mc<sup>2</sup>.
Here is another example of the use of this formula in physics calculations. Recently there has been quite a controversy over whether neutrinos were observed traveling at a speed faster than light. Relativity doesn't allow that, and, since neutrinos have nonzero (but incredibly tiny) mass, they aren't even supposed to travel ''at'' the speed of light. This very issue came up on the [[Talk:Main_Page#Neutrinos]]. The speeds under discussion were calculated by the use of E=mc<sup>2</sup>.
Several scientists have gone on record stating that the neutrinos, which have mass, travel at precisely the speed of light. If true, this disproves the [[Theory of Relativity]] and the claim that '''E=mc<sup>2</sup>'''. However, it is more likely that those scientists are using language inaccurately. It is impossible to measure the speed of neutrinos precisely. What is meant is the difference between the speed of light and the speed of the neutrinos is too small to measure.
Several scientists have gone on record stating that the neutrinos, which have mass, travel at precisely the speed of light. If true, this disproves the [[Theory of Relativity]] and the claim that '''E=mc<sup>2</sup>'''. However, it is more likely that those scientists are using language inaccurately. It is impossible to measure the speed of neutrinos precisely. What is meant is the difference between the speed of light and the speed of the neutrinos is too small to measure.
==Deducing the Equation From Empirical Observation==
While the equation was historically developed on theoretical grounds as an inevitable consequence of special relativity, it is possible to deduce it purely from empirical observation.
So, for the purposes of this section, imagine that one is in the era of "classical physics"; prior to 1900 or so. Relativity has not been invented, but, inexplicably, nuclear physics has. Imagine that the phenomena of radioactivity and nuclear fission have been observed, without any knowledge of relativity.
A well-accepted physical law of classical physics was the law of conservation of mass. This was not easy to deduce. It required careful analysis of such phenomena as combustion, in the 1700's, to eliminate the various confounding sub-phenomena that made the law difficult to see. But, by 1900, the law was well established:
:::*'''In all interactions, mass is precisely conserved.'''
For example, the mass of a TNT molecule is 227.1311 Daltons, or 227.1311 g/mol, which is, for all practical purposes, the same as the mass of its constituent Carbon, Hydrogen, Nitrogen, and Oxygen atoms. It is essentially impossible to measure the difference. The principle of conservation of mass is upheld.
But when nuclear phenomena are discovered, we notice something different. The masses of the result particles after an event (e.g. alpha decay, nuclear fission, or artificial transmutation) is measurably less than the masses of the original particle(s). With the invention of the mass spectrometer around 1920, it became possible to measure atomic weights of various isotopes with great precision.
Radium-226 decays into Radon-222 by emission of an alpha particle with an energy of 4.78 MeV.
1 kg of Radium-226 = <math>\frac{.622 \times 10^{27}}{226.0254}</math> atoms. (The numerator is Avogadro's number, and the denominator is the atomic weight of Radium-226.) This is 2.6643647 * 10<sup>24</sup> atoms.
That number of Radon-222 atoms has mass .98226836 kg. That number of alpha particles has mass .01770863 kg.
The mass lost is .00002301 kg.
Each emitted alpha particle has energy of 4.78 MeV, or 4.78 * .1602 * 10<sup>-18</sup> Joules. The total alpha energy from the decay of 1 kg of radium is 2.04 * 10<sup>12</sup> Joules.
Also, Radon-222 decays into Polonium-218 by emission of an alpha particle with an energy of 5.49 MeV.
1 kg of Radon-222 = <math>\frac{.622 \times 10^{27}}{222.0176}</math> atoms. This is 2.7124611 * 10<sup>24</sup> atoms.
That number of Polonium-218 atoms has mass .98194467 kg. That number of alpha particles has mass .01802830 kg.
The mass lost is .00002703 kg.
Each emitted alpha particle has energy of 5.49 MeV. The total alpha energy from the decay of 1 kg of polonium is 2.39 * 10<sup>12</sup> Joules.
It looks as thought we have to rewrite the law of conservation of mass:
:::*'''In all "ordinary" interactions, mass is precisely conserved.'''
:::*'''In nuclear interactions, there is a small but measurable loss of mass.'''
:By the way, we can clearly see that atomic weights of pure isotopes are not integers, and that it has something to do with the energy released by nuclear disintegration. In retrospect, the formula E=mc^2 explains the non-integer character of atomic weights.
Making special cases like this is unsatisfactory, of course.
We do this for a few other interactions, including the explosion of TNT. This would include the Lithium-plus-Hydrogen and Uranium fission phenomena described above. We won't bother with the details. As observational scientists, we look for patterns in the behavior of nature. We make a table:
{| class="wikitable" border="1" cellpadding="8" cellspacing="0"
! interaction
! energy released per kg, Joules
! mass lost per kg of original substance, kg
|-
| explosion of TNT
| 4.184 * 10<sup>6</sup>
| seems to be zero
|-
| alpha decay of Ra-226
| 2.04 * 10<sup>12</sup>
| .00002301 kg
|-
| alpha decay of Rn-222
| 2.39 * 10<sup>12</sup>
| .00002703 kg
|}
We plot these, and a few others, not shown, on graph paper, and find to our amazement that the relationship is linear.
For Ra, m/E = .112794118 E-16
For Po, m/E = .113096234 E-16
If this is linear, the mass defect for TNT would have been .47 * 10<sup>-10</sup>. We couldn't possibly have measured this.
So we can rewrite the rule for conservation of mass in a more satisfactory way:
:::*'''In all interactions, there is a loss of mass, equal to about .113 * 10<sup>-16</sup> kg per Joule of energy released.'''
What we thought was exact conservation is just very nearly exact, and we hadn't been able to measure it before.
But maybe there's more. This constant has dimensions of kilograms per Joule. From high-school physics, we know that that is seconds squared divided by meters squared. That is, it is the reciprocal of the square of a velocity. We calculate that velocity. It is about 2.97 * 10<sup>8</sup> meters per second. Very close to the speed of light! Very interesting! (The calculations above were not extremely precise. The formula has been verified with great precision, but not here.)
We don't understand why (that will have to wait for the invention of relativity), but we can formulate a hypothesis:
:::*'''In all interactions, there is a loss of mass, equal to <math>\frac{1}{c^2}</math> times the amount of energy released.'''
We don't have to give the units any more, since everything is now dimensionally correct.
::There is a very interesting analogy with the discovery of [[Maxwell's equations]]. Maxwell found an interesting relationship involving the fundamental constants <math>\epsilon\,</math> and <math>\mu\,</math> appearing in his equations. Specifically, <math>\epsilon\mu\,</math> has the dimensions of seconds squared divided by meters squared, and that:
:::::<math>\frac{1}{\epsilon\mu} = c^2</math>
::where "c" was the known velocity of light. He also showed that his equations predict electromagnetic waves, propagating at that speed.
This is a common way that scientific discoveries are made. But, as stated above, this is not how E=mc^2 was discovered. But it could have been.
==See also==
==See also==
[[Counterexamples to Relativity]]
[[Counterexamples to Relativity]]
[[essay:Rebuttal to Counterexamples to Relativity]]
== References ==
== References ==